Assessment of Bone Quality from pQCT Images Dean Inglis, Ph.D. Assistant professor (adjunct)...
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Transcript of Assessment of Bone Quality from pQCT Images Dean Inglis, Ph.D. Assistant professor (adjunct)...
Assessment of Bone Quality from pQCT Images
Dean Inglis, Ph.D.
Assistant professor (adjunct)
Department of Civil Engineering
McMaster University
Overview
CT image source, formation and characteristics Image segmentation Bone morphometry 2D stereology: basic principles, assumptions 3D stereology: mean intercept lengths, Eigen
analysis, interpretation Model independent measures Topology: Euler number, Structure Model Index Summary
What is Peripheral Computed Tomography? pQCT (2D), hr-pQCT (3D) CT imaging techniques that target
peripheral sites use computer controlled X-ray source +
detector system multiple X-ray 1D/2D projections
reconstructed into 2D slice/3D volume images
spectrum
CT basic principles
electron beam strikes tungsten target and generates polychromatic X-ray beam
source
CT basic principles
X-rays pass through a sample and are attenuated:
I = Ioe - ∫ u(x,y) ds
I = intensity at the detector Io= intensity of the source u(x,y) = attenuation characteristics of the
sample: depend on atomic number (density) attenuation is integrated along a ray
CT basic principles
emergent X-rays detected by a phosphor detector coupled to a CCD camera
CT image formation
detection of many rays results in a projection (silhouette) of the sample
many projections are generated by rotating the source and detector around the sample
image is reconstructed using convolution back-projection
CT image formation
CT image formation
CT image characteristics
raw CT data represent linear attenuation coefficients
coefficients are converted to CT numbers, Hounsfield Units (HU), in the reconstruction process
pQCT calibrates HU into density: g/cm3
Image characteristics
an image in its most basic sense is a matrix of numbers a 2D matrix has topology consisting of pixels (picture
elements) 8-connected to their neighbours images have a spatial origin, eg. (0,0,0) mm, and finite
spacing between their pixel centers, eg. 0.5×0.5×0.5 mm3
spacing partly governs ability to resolve small features accurately
pQCT resolution: 0.2×0.2×0.5 mm3 (non-isotropic) hr-pQCT resolution: 0.08×0.08×0.08 mm3 (isotropic)
Topology example: 6x5 image
xi,yi
5
1
2
3
4
6
7 8
Image characteristics
a 3D image can be considered as a stack of 2D images having thickness
pixels are now called voxels (volume elements) and are 27-connected topologically
Image segmentation
segmentation is the task of classifying pixels/voxels based on their value and topological affinity
segmentation is used to isolate features of interest (bone) in an image
Image segmentation
Image segmentation
thresholding:
P(x,y,z) = Po(x,y,z) < t ? 0 : Po(x,y,z)
Image segmentation
binarization:
P(x,y,z) = Po(x,y,z) < t ? 0 : 1
Image segmentation
some problems to consider… how do we pick “t” without bias? how do we pick one bone from another? how do we pick bone constituents
(cortex vs trabeculae)?
Image segmentation
bone images consist of 2 pixel groups: bone and soft tissue (or background): a histogram of a bone image appears bimodal
segment bone from non-bone using an automated thresholding scheme to determine “t”
Otsu’s method minimizes the error of misclassifying a non-bone pixel as bone and vice versa by minimizing the within-class variance of the two groups
Otsu : t
Image segmentation at low resolution Otsu fails for bone within
bone: cortical bone vs. trabecular bone trabecular bone vs. marrow
Image segmentation
many other schemes exist: livewire tracing, active contours, level
sets desirable characteristics of any method: simple, fast, reproducible, automated,
gets the job done!
Bone morphometry given a segmented image of bone,
what can be measured? HU’s represent attenuation: analog for
density calibration allows volumetric BMD
(g/cm3): BMD = ∑ [Pi != 0 ? m×Pi + b : 0 ]
segmentation provides volume (cm3):V = [ ∑ Pi != 0 ? 1 : 0 ]×dx×dy×dz
BMC = BMD × V (g)
Bone morphometry
what is structure and is it important? 3 plank beam: σ = My/I I-beam / block ~ 4 for L / t = 5 in addition to density (stiffness), the
spatial arrangement of material
(structure) contributes to strength BMD/BMC is limited:
no information on spatial arrangement
Bone morphometry
how can structure be measured? before CT, samples were embedded in resin,
sliced and polished, and photomicrographed 2D images: area, perimeter length, number more information (e.g., thickness, spacing)
can be inferred using stereology: mathematical science based on geometric probability
2D stereology
Parfitt et. al. developed the “parallel plate model” for analyzing 2D images
(J. Clin. Invest. 1983, v72, 1396-1409) key assumptions:
-trabecular bone comprised mainly of interconnected plates
-tissue is isotropic
-sample is uniformly randomly obtained
2D stereology
basic 2D quantities:
PB = bone perimeter length (mm)
AB = bone area (mm2)
AT = tissue section area (mm2) (bone + marrow)
2D stereology
bone volume fraction (%):
TBV = BV/TV = AB / AT
Bone surface density (mm2/mm3):
Sv = BS/TV = PB / AT
bone surface to volume ratio (mm2/mm3):
S/V = BS/BV = PB / AB
mean trabecular plate thickness (mm):
MTPT = Tb.Th = 2 AB / PB
mean trabecular plate density (/mm):
MTPD = Tb.N = BV/TV / Tb.Th = PB / (2 AT) mean trabecular plate separation (mm):
MTPS = Tb.Sp = 1 / Tb.N – Tb.Th = 2 (AT – AB) / PB
3D stereology
trabecular bone is a highly
organized 3D oriented structure 3D provides additional metrics:
surface area, volume, orientation a stereologic technique using a 3D
array of line probes provides BV/TV, Tb.Th, Tb.N, and Tb.Sp
3D stereology
considering the 2D case, focus on the boundary between bone and marrow within a circular ROI
overlay an array of test lines spaced δ apart
the sum of test line lengths, L, is orientation independent
this is only true with uniform sampling: circular ROI
3D stereology
consider the intercepts between test lines and boundaries
the number of intercepts, Tb.N(θ), depends on orientation
the sum of intercept lengths, ∑I, is orientation independent as δ→0
BV/TV = ∑I / L mean intercept length, a.k.a. Tb.Th:
MIL(θ) = ∑I / Tb.N(θ) the number of intercepts in marrow,
M.N(θ), is not equal to Tb.N(θ) Tb.Sp(θ) = ( L - ∑I ) / M.N(θ)
3D stereology
in 2D, an ellipse can be fit to data from N orientations
Let (xi, yi) = (cos(θi), sin(θi)), i = 1→N Tb.N(θi) = A xi
2 + B xiyi + Cyi2
least squares fitting gives A,B and C arranging A, B, C into a 2×2 matrix:
A ½B ½B C Eigen analysis gives the orientation and
lengths of the principle axes of the ellipse
anisotropy is defined as the ratio of the axes’ lengths: L2 / L1
x
y
θL1L2 L1L2 L1L2
3D stereology
in 3D, a 3D array of parallel test lines probes the image uniformly within a spherical ROI
“uniformly” means equal area partitions of the surface of a unit sphere or many random orientations
orientation of the lines is defined in terms of two angles: θ, φ
( xi, yi, zi ) = ( sin(θi)cos(φi), sin(θi) sin(φi), cos(θi) )
Tb.N( θi, φi ) = A xi2 + B yi
2 + C zi2 + D
xiyi + E xizi + F yizi
θ
φx
y
z
3D stereology
least squares fitting gives A,B,C,D,E,F A,B,C,D,E,F are arranged to form a 3×3 matrix Eigen analysis gives the orientation and
lengths of the 3 principle axes of the ellipsoid anisotropy is defined by the ratios of the axes’
min to max lengths: L3 / L1, L2 / L1
L2
L3
L1
y
z
x
Model independent measures
Tb.Th and Tb.Sp can be measured without model assumptions
find the medial axes (2D) or surface (3D) of the bone (marrow)
fit maximal non-overlapping spheres within bone (marrow)
analyze the histogram of spherical diameters
works for any ROI shape
Topology
the Euler Number is an index of connectivity of trabecular bone
measures redundant connectivity: the degree to which parts of the object are multiply connected:Χ = β0 – β1 – β2
β0 is the number of isolated objects = 1 for bone
β1 is the connectivity β2 is the number of enclosed cavities = 0 for
bone β1 is calculated by analyzing the local
neighbourhood connectivity of each voxel representing bone
works for any ROI shape
Topology
the Structure Model Index, SMI, is a measure of the degree of convexity of a structure
in bone, it indicates the relative prevalence of rods and plates
SMI is calculated by differential analysis of the triangulated surface of the bone:SMI = 6 BV ( dBS/dr ) / BS2
dBS/dr is estimated by translating the surface by a small distance, dr, in its normal direction: dBS/dr = (S´ - S) / dr
an ideal plate, cylinder (rod) and sphere have SMI values of 0, 3, and 4
Topology
a shell… and its inflated surface transition of a rod to a
plate… perforation of a plate…
h:r = 10, SMI = 2.97h:r = 5, SMI = 3.02h:r = 1, SMI = 2.61h:r = 0.5, SMI = 2.00h:r = 0.05, SMI = 0.35r:R = 0, SMI = 0.35r:R = 0.05, SMI = 0.39r:R = 0.25, SMI = 0.49r:R = 0.5, SMI = 0.69r:R = 0.75, SMI = 1.16r:R = 0.87, SMI = 1.70r:R = 0.95, SMI = 2.09
Summary
pQCT is an X-ray tomographic imaging modality
pQCT provides high resolution 2D / 3D images
images of trabecular (and cortical) bone can be digitally partitioned into bone/non-bone
bone (quality) can be numerically characterized in terms of BMD and structure
structure can be quantified using stereological and topological methods
stereological methods may have embedded assumptions / limitations
model independent measures
Finis!
further reading:http://www.scanco.ch/support/general-
faq.html#c781http://www.stratec-med.com/en/
prod_xct2000.php